Table Of Links
Abstract
- Acknowledgements & Introduction
2. Universal properties
3. Products in practice
4. Universal properties in algebraic geometry
5. The problem with Grothendieck’s use of equality.
6. More on “canonical” maps
7. Canonical isomorphisms in more advanced mathematics
8. Summary And References
Products in Practice
When a mathematician writes X × Y , what do they mean? Is it a product in the sense of the universal property, or is it the “special” one X × Y consisting of ordered pairs? One might imagine that, to fix our ideas, it’s easiest to just choose the special one. On the other hand, a mathematician would almost certainly agree with the following claim
R 2 × R = R × R 2 = R 3 ;
It is as clear as the claim that 2 + 1 = 1 + 2 = 3. However, it seems to be impossible to set up the foundations of mathematics in such a way that all of these sets are literally equal. Using the model of products in the previous section, a typical element of R 2 × R looks like ((a, b), c) and a typical element of R × R 2 looks like (a,(b, c)). These two constructions clearly carry the same data, and yet equally clearly they are not identical; they are both different models for R 3 , as is the model consisting of ordered triples (a, b, c) defined for example as functions {1, 2, 3} → R. In particular, sets equipped with the product do not strictly speaking form a monoid (because (A × B) × C = A × (B × C) is strictly speaking false).
However all three of R 2 × R, R × R 2 and R 3 satisfy the universal property for a product of three copies of R, meaning that there are unique isomorphisms between these constructions. The category theorists would tell us that the category of sets equipped with the product can be made into a monoidal category, which means that we can write down the extra data of a collection of isomorphisms iABC : (A × B) × C ∼= A × (B × C) satisfying an equation called the pentagon axiom [Wik04a], which says that the two resulting natural ways of identifying ((A × B) × C) × D with A × (B × (C × D)) are equal. Unsurprisingly, in this example, both of the natural identifications send (((a, b), c), d) to (a,(b,(c, d))).
It is axioms like the pentagon axiom – “higher compatibitilies” between identifications of objects which mathematicians are prone to regard as equal anyway – which are so easy to forget. Which of ((A×B)×C)×D and A×(B×(C ×D)) does a mathematician mean when they write A × B × C × D? If one (strictly speaking, incorrectly) decides that the sets ((A × B) × C) × D and A × (B × (C × D))) are equal it doesn’t matter! There is only one way in which two sets can be equal (in contrast to there being many ways of being isomorphic, in general), and if we think this way then we deduce the pentagon axiom no longer needs to be checked! It is phenomena like this which gives rise to arguments which are strictly speaking incomplete, throughout the literature. Note of course that in every case known to the author, these arguments can be filled in; however the Lean community has only just started on algebraic geometry, and it will be interesting to see what happens as we progress.
I have mentioned the real numbers already. They are unique up to unique isomorphism, and mathematicians do a very good job of sticking to the universal property and developing calculus using only the completeness property of the reals rather than relying on any kind of explicit set-theoretic definition. When it comes to products however, we don’t to this. Consider for example φ : R 2 → R defined by φ(x, y) = y 2 + xy − x.
Mathematicians would have no objection to that definition – however it assumes the ordered pair model for the reals: it is a function from the product rather than from a product. If (P, π1, π2) is a product then we can define φP on P by φP (t) = π2(t) 2 + π1(t)π2(t) − π1(t). This looks rather more ungainly than the definition of φ above so is typically avoided. However, if one wants to identify sets like (A × B) × C and A × (B × C) on the basis that there is a unique isomorphism between them satisfying various basic properties, then one is strictly speaking forced to develop a theory of products of sets using only the universal property.
:::info
Author: KEVIN BUZZARD
:::
:::info
This paper is available on arxiv under CC BY 4.0 DEED license.
:::
